# Cluster Partition Function and Invariants of 3-manifolds

## TL;DR

This paper reviews recent advances in Chern-Simons theory on hyperbolic 3-manifolds, introducing the cluster partition function as a novel computational tool leveraging cluster algebra techniques, with applications in string theory.

## Contribution

Introduction of the cluster partition function as a new method for evaluating Chern-Simons path integrals on knot complements in hyperbolic 3-manifolds.

## Key findings

- Cluster partition function effectively computes Chern-Simons invariants.
- Connections established between cluster algebra techniques and topological quantum field theory.
- Open questions and potential applications in string theory are discussed.

## Abstract

We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold $M$ with complex gauge group $G$. We focus on the case $G=SL(N,\mathbb{C})$ and with $M$ a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory.